Lipschitzian subspaces in Clifford algebras

In every Clifford algebra Cℓ(V,q) there is a Lipschitz monoid Lip(V,q) which in general is the multiplicative monoid (or semi-group) generated by V in Cℓ(V,q); its even and odd components are closed irreducible algebraic submanifolds. When dim(V)⩽3, they are trivially the even and odd components of Cℓ(V,q). When dim(V)⩾4, it is sensible to search for all linear subspaces contained in Lip(V,q) if we wish better to know the geometry of these algebraic submanifolds. All these lipschitzian subspaces are described here, and many properties are established which involve the essential concept of “adjacent lipschitzian elements”. In particular there are two families of maximal lipschitzian subspaces; the regular ones have the same dimension as V, and they are in bijection with the lipschitzian lines; the singular ones (which are not adjacent to anything but 0) have dimension 4, and their study involves Vahlen matrices.